There is no known proof that no cycle, other than the rather trivial 4-2-1
exists in positive integers. However it is known that no such cycle can exist
with a 'small' number of elements (we'll be more exact later on).

The current status on 3x+1 cycles is based on a theorem by Crandall.
In 1978 he proved the following:

Theorem (Crandall, 1978)

Define 3x+1 trajectories using (3x+1)/2 for odd elements and x/2 for even elements.
Let N0 be the lowest element of a 3x+1 cycle
in positive integers of cycle length k.
Furthermore, let pj / qj be a convergent with j > 4
of the expansion of the continued fraction of ln(3) / ln(2).
Then k > (3/2) * min ( qj , 2 * N0 / (qj + qj+1)

To elaborate the practical consequences of this theorem the first task is therefore
to work out the convergents of the continued fraction of ln(3) / ln(2).
There is apparently no 'neat' way of obtaining these factors so these have to be
calculated numerically. With some labor the following table of convergents can be found:

j

pj

qj

1

1

1

2

2

1

3

3

2

4

8

5

5

19

12

6

65

41

7

84

53

8

485

306

9

1,054

665

10

24,727

15,601

11

50,508

31,867

12

125,743

79,335

13

176,251

111,202

14

301,994

190,537

15

16,785,921

10,590,737

16

17,087,915

10,781,274

17

85,137,581

53,715,833

18

272,500,658

171,928,773

19

357,638,239

225,644,606

20

630,138,897

397,573,379

21

9,809,721,694

6,189,245,291

22

10,439,860,591

6,586,818,670

23

103,768,467,013

65,470,613,321

From this table it is possible to determine a minimal cycle length
once it is known that all numbers below a particular N actually are convergent
(eventually reach 1).

Example :

Assume that all numbers up to 1500 can be shown to converge to 1.
Therefore no cycle (other than 4-2-1) can have a lowest element < 1500.
Assume for the moment that a cycle exists with 1500 as its lowest element.
Now from the table above take j = 6. We see that q6 = 41.
The factor 2 * N0 / (qj + qj+1)
works out as 2 * 1500 / (41+53) which is roughly 31.
Since 31 < 41 we find that the cycle length can not be lower
than 3/2 * 31 = 47.

The example above clearly shows two things. First of all the minimal cycle length
can be determined from the lowest N that is not yet known to be convergent.
Secondly the minimal cycle length does not increase gradually with N, but
increases in intervals, depending on the values of qj.

For every j there is a maximal critical value (Nmax) of N
beyond which higher values of N do not give any further improvement of the
minimal cycle length. The value of Nmax is equal to
qj * (qj + qj+1) / 2 .
For j = 6 this value lies at N = 1927. At this value
q6 = 2 * N0 / (q6 + q7)
so both expressions of the theorem yield identical numbers. Once it is known
all numbers below 1927 are convergent the minimal cycle length can therefore
be set at (3/2) * 41 which yields 62.

To increase the minimal cycle length beyond this value the next row at j=7 is needed.
For lower N though the value of the second expression is below 41 and the minimal
cycle length can therefore not be increased. It is only when N reaches a
minimal critical value (Nmin) for this row that further
improvements can be obtained.
The value of Nmin can be seen to lie at
qj-1 * (qj + qj+1) / 2 .
For j=7 this value lies at N = 7360. At this value
2 * N0 / (q7 + q8)
is equal to 41, and higher N therefore yield higher values for the minimal cycle length.

Working out these critical values for rows with j > 4 yields the following values :
(Note : Bigger entries for Nmin and Nmax are rounded)

j

qj

Nmin

Nmax

kmin at Nmax

1

1

2

1

3

2

4

5

5

12

132

318

18

6

41

564

1,927

62

7

53

7,360

9,513

80

8

306

25,732

148,563

459

9

665

2,488,698

5,408,445

998

10

15,601

15,783,110

370,274,134

23,402

11

31,867

867,431,201

1,771,837,067

47,801

12

79,335

3,035,921,290

7,558,126,447

119,003

13

111,202

11,969,231,783

16,776,990,139

166,803

14

190,537

599,449,615,674

1,027,115,802,069

285,806

15

10,590,737

2,036,079,429,954

113,172,673,831,054

15,886,106

16

10,781,274

341,535,948,748,930

347,680,491,387,159

16,171,911

17

53,715,833

1,216,368,161,954,000

6,060,343,986,623,000

80,573,750

18

171,928,773

10,677,992,615,805,000

34,177,151,614,467,000

257,893,160

19

225,644,606

53,574,551,736,291,000

70,312,888,338,719,500

338,466,909

20

397,573,379

743,140,051,792,797,000

1,309,718,777,460,900,000

596,360,069

21

6,189,245,291

2,539,711,459,647,450,000

39,537,096,854,067,000,000

9,283,867,937

22

6,586,818,670

222,990,560,815,925,000,000

237,314,619,175,287,000,000

9,880,228,005

23

65,470,613,321

The results clearly show that the minimal possible cycle length is currently
pretty large. This is because current results almost certainly indicate
that all numbers below 500 . 1015 are convergent.
Up to this limit all numbers were checked for convergence independently both by
Tomás Oliveira e Silva
and the author (see current status),
using different software and running on different hardware platforms. The
'almost certainly' simply references the fact that it is almost impossible to
guarantee one hundred percent correctness when one makes computer algorithms run
for many CPU years. The author firmly believes though that the results are correct.
Note also that no inconsistencies between the two result sets were encountered.

If we accept therefore that all numbers below
500 . 1015 are indeed convergent
then the numbers in row 19 indicate that a 3x+1 cycle must contain at least
338,466,909 (or roughly 330 million) elements. Note that this is likely to
remain the lower limit for a bit longer, since it can only be improved by
demonstrating the convergence of all numbers below 743 . 1015,
which will take more computing effort.

Finally we should take into account that the values above are valid for
the variety of the 3x + 1 problem where an 'odd' iteration is defined as
resulting in (3x+1) / 2. If one defines such an iteration as simply 3x+1,
cycle lengths increase by a factor ln(6) / ln(3) ≈ 1.63.
Using this definition the minimal length for a non-trivial cycle is
currently roughly 553 million.

Reference :

The proof of Crandall's Theorem can be found in
R. E. Crandall, On the "3x+1" problem, Math Comp, 32 (1978) 1281-1292.

Acknowledgement :

Many thanks to Vic Vyssotski for working out all the convergents
of the continued fraction as depicted in the first table and
supplying helpful insights as well.